On the planar inertial motion of a system of three coupled bodies

Numerous papers are devoted to studying the motion of a system (coupling) of two bodies in the Earth’s satellite orbit ([1–4] and others). The problem on the planar inertial motion of three bodies, coupled by a non-extensible weightless string having the form of an unfastened chain, is considered in the paper. Such a configuration can be represented, for example, by a system of two coupled spacecraft rotating around their common center of mass (in order to simulate the gravity force) in long-term space missions, when the third body (the lift) is located on a connecting cable. The bodies are considered to be the material points (particles).     

Symmetric periodic solutions of the Hill’s problem. I

The planar circular Hill’s problem is considered, as well as its limiting integrable variant called the Hénon problem, for which the original Hill’s problem is a singular perturbation. Among solutions to the Hénon problem there are a countable number of generating solutions-arcs that are uniquely determined by the condition of successive passage through the origin of coordinates—singular point of equations of motion of the Hill’s problem. Using the generating solutions-arcs as “letters” of a certain “alphabet”, one can compose, according to some rules, the “words”: generating solutions of families of periodic orbits of the Hill’s problem. The sequence of letters in a word determines the order of orbit transfer from one invariant manifold to another, while the set of all properly specified words determine the system’s symbolic dynamics.     

Optimization of passive orbit with the use of gravity maneuver

A high-precision method of calculating gravitational interactions is applied in order to determine optimal trajectories. A number of problems, necessary for determination of optimal parameters at a launch of a spacecraft and during its flyby near celestial bodies, are considered. The spacecraft trajectory was determined by numerical integration of the equations of passive motion of the spacecraft and of the equations of motion for planets, the Sun, and the Moon. The optimal trajectory of the spacecraft approaching the Sun is determined by fitting its initial conditions.     

Stabilization of a Reentry Vehicle by a Partial Spin-up during Uncontrolled Descent

Stabilization of a reentry vehicle (RV) by a partial spin-up of it is considered for the case of uncontrolled descent into the atmosphere. In this case, the vehicle is a composite construction consisting of two rigid bodies, a return capsule and a stabilizing block, which is put in rotation. A model is developed for the spatial motion of the reentry vehicle considered as a system of coaxial rigid bodies rotating about a common axis of symmetry. The free motion is studied, and the stability of steady-state regimes is analyzed. The spatial motion of the system is considered for the case of a small asymmetry due to displacement of the axes of dynamic symmetry of the bodies with respect to the spin axis, and approximate solutions for the motion parameters of the free system are found.     

On analysis of close encounters of two cosmic bodies in near almost-circular orbits

This paper is devoted to the study of relative motion and close encounters of two cosmic bodies located in near almost-circular orbits. This problem is topical due to the asteroid hazard proceeding from the NEA group asteroids located in the near-Earth orbits. The (99942) Apophis asteroid, a representative of this group discovered in July 2004 by the Kitt Peak observatory (Arizona), is considered as an example.     

Kinematics of relative motion of two space bodies in close almost circular orbits

The paper is dedicated to a qualitative investigation of relative motion and close convergences of two space bodies located in close almost circular orbits. This problem is topical due to the asteroid hazard originating from the NEA group asteroids located in the orbits close to that of the Earth. P.E. El’yasberg [1] considered similar problems in the 1960s in relation to Earth’s artificial satellites in close almost circular orbits.     

The family of potentials admitting integration of the perturbed two-body problem in regular coordinates

An infinite system of potentials is presented that admits separation of regular variables in the perturbed two-body problem. The regular coordinates are constructed using a specially selected L-matrix. An explicit solution to the problem in the elliptical case is constructed. In the general case the solution is reduced to inversion of hyper-elliptic integrals. The cases of motion with and without constraints are considered. The results of numerical experiments are presented.     

Design of satellite constellations with continuous coverage on elliptic orbits of <Emphasis Type="Italic">Molniya</Emphasis> type

New methods of choosing the structures of satellite constellations (SC) on elliptical orbits of the Molniya type are presented. The methods, using critical inclination and putting the orbit apogee in the Earth’s hemisphere with an area of continuous coverage, are based on geometrical analysis of two-dimensional representation of the coverage conditions and SC motion in the space of inertial longitude of the orbit ascending node and time. The coverage conditions are represented in the form of a certain region. Dynamics of all satellites in this space is represented by uniform motion along a straight line approximately parallel to the ordinate axis, while the satellite system forms a grid. The problem of choosing a minimal (as far as the number of satellites is concerned) SC configuration can be formulated as a search for the most sparse grid. The contemporary advanced methods of computational geometry serve as an algorithmic basis for the problem solution. Design of SC for continuous coverage of latitude belts with the use of kinematically regular systems is considered. A method of analyzing single-track systems for continuous coverage of arbitrary geographic regions is described, which makes a region at any time instant observable by at least one satellite of the system. As an example, SC on elliptical orbits are considered with periods of ～4, 12, and 24 hours.     

Analysis of the dynamics of the deployed aerodynamic space tether system

An analysis of the motion of a deployed space system that consists of two end bodies connected by a tether has been considered. One of the bodies has a relatively large ballistic coefficient that ensures aerodynamic braking or the stabilization of the motion of the entire system in relatively low near-Earth orbits. The deployment of this system mainly occurs due to the action of aerodynamic forces. Several ways of deploying the system have been analyzed, including (1) the uncontrolled release of the tether with hardly any braking; (2) deployment with constant braking force; (3) the dynamic control law without feedback, when the resistance force varies according to a set program; (4) a kinematic control law with feedback when programs are set for varying the velocity and length of the tether release. To analyze the dynamics of the system, a mathematical model of motion has been constructed in which the motion of the end bodies relative to their centers of mass is taken into account.     

Evolution of motion of two viscoelastic planets in the field of forces of their mutual attraction

Translational-rotational motion of two viscoelastic planets in a gravitational force field is studied. The planets are modeled by homogeneous isotropic viscoelastic bodies. In their natural undeformed state each of the planets represents a sphere. We investigate a specific case when the planet’s centers of mass move in a fixed plane, the axis of rotation for each planet being directed along the normal to this plane. An equation describing the evolution of a slow angular variable (perihelion longitude) is derived. The observed displacement of the perihelion of Mercury is compared with the results obtained in the considered model problem about motion of two viscoelastic planets. Quite important is the fact that the planet of smaller mass (Mercury) moves not in a central Newtonian field of forces, but rather in the gravitational field of a rotating viscoelastic planet (Sun).     

Transition of Planar Rotational Motion of an Asymmetric Spacecraft into Oscillatory Motion at Its Reentry into the Atmosphere

The motion of a spacecraft with small asymmetry relative to its center of mass is considered. The restoring aerodynamic moment of the spacecraft is described by the Fourier series in terms of the angle of attack with the two first sinusoidal and the first cosinusoidal terms. A solution for the angle of attack in the undisturbed rotational motion is found. The analytical expression is obtained for the integral of action taken along the separatrices that separate the rotational and oscillatory regions of the phase portrait of a system. The transition of the spacecraft's motion from planar rotational to oscillatory is investigated. This transition is caused by a slow variation of moment characteristic coefficients, as well as by the presence of small asymmetry and damping and slow variation of their coefficients. Analytical formulas are obtained for determining the times of transition from rotational to oscillatory motion, as well as for the critical angular velocity of beyond-the-atmosphere rotation. When this critical velocity is exceeded, body rotation proceeds for a long time interval (planar autorotation arises).     

Consequences of collisions of natural cosmic bodies with the Earth’s atmosphere and surface

Possible consequences of collisions of natural cosmic bodies with the Earth’s atmosphere and surface are described. The methodological basis of classification of consequences is the solution of meteor physics equations characterizing the trajectory of a body in the atmosphere, namely, the dependence of the body’s velocity and mass on the flight altitude. The solution depends on two dimensionless parameters characterizing the drag altitude and the role of mass loss by a meteoroid during its motion in the atmosphere. Depending on values of these parameters, the degree of effect on the planetary surface considerably changes. In particular, the conditions of cratering and meteorite fall on the planetary surface are obtained. The results are presented in a simple analytical form. They quite match to the real events considered in the paper. Recommendations are given on further investigations into the important problem of interaction of cosmic bodies with planetary atmospheres.     

Ballistics of a fragmenting meteor body with allowance made for ablation in the non-isothermal atmosphere. II

The second part of the work is devoted to studying the motion of fragmenting meteor bodies with ablation by fragments in a planet’s atmosphere, taking into account its non-isothermal character in altitude. The mechanical fragmentation of a meteoroid under the action of aerodynamic drag is considered in the context of statistical strength theory. An analytical solution for ballistics of a fragmenting body is obtained at a constant parameter of ablation. A theoretical analysis of the derived regularities is made and an example of their usage in the problem of a comet-asteroid hazard is presented.     

空间机械臂建模方法与控制策略研究

针对刚柔耦合空间机械臂动力学建模中对柔性体采用的传统描述方法（有限元法、模态综合法以及集中参数法等）并不足以精确描述柔性大变形的问题,采用绝对节点坐标法描述柔性体,采用自然坐标法描述刚性体,建立了末端带集中质量的双连杆柔性机械臂的动力学模型并且研究了机械臂的空间定位问题。结合广义α法以及工程上常用的Scaling技术,开发了计算程序,实现了动力学方程的高效精确数值求解。针对机械臂的空间定位以及柔性变形问题,提出了一种运动规划方案,采用PD控制策略,实现了机械臂的运动跟踪控制;仿真结果表明：提出的运动规划方案能有效地减弱机械臂的柔性变形。     

Families of periodic solutions in three-dimensional restricted three-body problem

Planar orbits of three-dimensional restricted circular three-body problem are considered as a special case of three-dimensional orbits, and the second-order monodromy matrices M (in coordinate z and velocity v _z ) are calculated for them. Semi-trace s of matrix M determines vertical stability of an orbit. If |s| ≤ 1, then transformation of the subspace (z, v _z ) in the neighborhood of solution for the period is reduced to deformation and a rotation through angle φ, cosφ = s. If the angle ? can be rationally expressed through 2π,φ = 2π·p/q, where p and q are integer, then a planar orbit generates the families of three-dimensional periodic solutions that have a period larger by a factor of q (second kind Poincareé periodic solutions). Directions of continuation in the subspace (z, v _z ) are determined by matrix M. If |s| < 1, we have two new families, while only one exists at resonances 1: 1 (s = 1) and 2: 1 (s = ?1). In the course of motion along the family of three-dimensional periodic solutions, a transition is possible from one family of planar solutions to another one, sometimes previously unknown family of planar solutions.     

Multi-structural estimation of an integrated navigation system of a reusable spacecraft

An approach to the synthesis of an integrated navigation system is considered for a reusable space-craft that performs an arbitrary spatial maneuver under the conditions of internal and external disturbances. The offered approach provides for a noise-suppressing solution of the navigation problem, both in a regular mode of spacecraft motion, and during its descent along the unplanned trajectory.     

General dynamics of a large class of flexible satellite systems

The paper presents a general formulation for librational dynamics of satellites with an arbitrary number, type and orientation of deploying flexible appendages. In particular, the case of beam-type flexible appendages deploying from a satellite in an arbitrary orbit is considered. The governing nonlinear, nonautonomous and coupled equations for vibration of the appendages and libration of the satellite are integrated numerically. Several cases of practical importance are considered making the system progressively more general and hence complex: (i) planar case representing pitch and appendage oscillations in the orbital plane; (ii) general attitude motion with planar vibrations of flexible members; (iii) above two cases together with the out-of-plane component of vibrations. Results show that under critical combinations of the system parameters the combined effect of flexibility and deployment can be substantial.     

Inductive interaction of electroconductive bodies with ionospheric and interplanetary plasma

The hypersonic hyperalfvèn plasma flow around electroconductive bodies is considered and electromagnetic inductive plasma-body interaction is analyzed. The possible applications of results obtained in this analysis to body motion in ionosphere and interplanetary plasma are discussed.     

The Motion of Coaxial Bodies of Varying Composition on the Active Leg of Descent

The motion of a spacecraft (SC) with double rotation and variable mass on the active leg of its descent is considered. The SC consists of two coaxial bodies. The coaxial scheme is used for gyroscopic stabilization of the SC longitudinal axis by the method of partial spin-up. The equations of spatial motion of coaxial bodies of varying composition are derived and approximate solutions for the angles of spatial orientation are found. The condition of decreasing amplitude of nutation oscillations is obtained, which allows the estimation of efficiency of the stabilization by partial spin-up. The errors in the magnitude and direction of the vector of braking thrust are also determined.__________Translated from Kosmicheskie Issledovaniya, Vol. 43, No. 3, 2005, pp. 224–232.Original Russian Text Copyright © 2005 by Aslanov, Doroshin, Kruglov.     

One Integrable Case of the Perturbed Two-Body Problem

The motion of a particle in a perturbed gravitational field of a point mass is considered. A constant force is taken as a perturbation. The L-matrix is selected in such a manner that in regular coordinates the separation of variables takes place. Integration of the equations of motion of the problem under consideration is carried out. The solutions are expressed in terms of elliptic functions. A qualitative investigation of the integrals of motion is made, and numerical examples are presented.